# Metrics in the sphere of a C*-module

Open Mathematics (2007)

• Volume: 5, Issue: 4, page 639-653
• ISSN: 2391-5455

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## Abstract

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Given a unital C*-algebra $𝒜$ and a right C*-module $𝒳$ over $𝒜$ , we consider the problem of finding short smooth curves in the sphere ${𝒮}_{𝒳}$ = x ∈ $𝒳$ : 〈x, x〉 = 1. Curves in ${𝒮}_{𝒳}$ are measured considering the Finsler metric which consists of the norm of $𝒳$ at each tangent space of ${𝒮}_{𝒳}$ . The initial value problem is solved, for the case when $𝒜$ is a von Neumann algebra and $𝒳$ is selfdual: for any element x 0 ∈ ${𝒮}_{𝒳}$ and any tangent vector ν at x 0, there exists a curve γ(t) = e tZ(x 0), Z ∈ ${ℒ}_{𝒜}\left(𝒳\right)$ , Z* = −Z and ∥Z∥ ≤ π, such that γ(0) = x 0 and $\stackrel{˙}{\gamma }$ (0) = ν, which is minimizing along its path for t ∈ [0, 1]. The existence of such Z is linked to the extension problem of selfadjoint operators. Such minimal curves need not be unique. Also we consider the boundary value problem: given x 0, x 1 ∈ ${𝒮}_{𝒳}$ , find a curve of minimal length which joins them. We give several partial answers to this question. For instance, let us denote by f 0 the selfadjoint projection I − x 0 ⊗ x 0, if the algebra f 0 ${ℒ}_{𝒜}\left(𝒳\right)$ f 0 is finite dimensional, then there exists a curve γ joining x 0 and x 1, which is minimizing along its path.

## How to cite

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Esteban Andruchow, and Alejandro Varela. "Metrics in the sphere of a C*-module." Open Mathematics 5.4 (2007): 639-653. <http://eudml.org/doc/269531>.

@article{EstebanAndruchow2007,
abstract = {Given a unital C*-algebra $\mathcal \{A\}$ and a right C*-module $\mathcal \{X\}$ over $\mathcal \{A\}$ , we consider the problem of finding short smooth curves in the sphere $\mathcal \{S\}\_\mathcal \{X\}$ = x ∈ $\mathcal \{X\}$ : 〈x, x〉 = 1. Curves in $\mathcal \{S\}\_\mathcal \{X\}$ are measured considering the Finsler metric which consists of the norm of $\mathcal \{X\}$ at each tangent space of $\mathcal \{S\}\_\mathcal \{X\}$ . The initial value problem is solved, for the case when $\mathcal \{A\}$ is a von Neumann algebra and $\mathcal \{X\}$ is selfdual: for any element x 0 ∈ $\mathcal \{S\}\_\mathcal \{X\}$ and any tangent vector ν at x 0, there exists a curve γ(t) = e tZ(x 0), Z ∈ $\mathcal \{L\}\_\mathcal \{A\} (\mathcal \{X\})$ , Z* = −Z and ∥Z∥ ≤ π, such that γ(0) = x 0 and $\dot\{\gamma \}$ (0) = ν, which is minimizing along its path for t ∈ [0, 1]. The existence of such Z is linked to the extension problem of selfadjoint operators. Such minimal curves need not be unique. Also we consider the boundary value problem: given x 0, x 1 ∈ $\mathcal \{S\}\_\mathcal \{X\}$ , find a curve of minimal length which joins them. We give several partial answers to this question. For instance, let us denote by f 0 the selfadjoint projection I − x 0 ⊗ x 0, if the algebra f 0 $\mathcal \{L\}\_\mathcal \{A\} (\mathcal \{X\})$ f 0 is finite dimensional, then there exists a curve γ joining x 0 and x 1, which is minimizing along its path.},
author = {Esteban Andruchow, Alejandro Varela},
journal = {Open Mathematics},
keywords = {C*-modules; spheres; geodesics; -modules},
language = {eng},
number = {4},
pages = {639-653},
title = {Metrics in the sphere of a C*-module},
url = {http://eudml.org/doc/269531},
volume = {5},
year = {2007},
}

TY - JOUR
AU - Esteban Andruchow
AU - Alejandro Varela
TI - Metrics in the sphere of a C*-module
JO - Open Mathematics
PY - 2007
VL - 5
IS - 4
SP - 639
EP - 653
AB - Given a unital C*-algebra $\mathcal {A}$ and a right C*-module $\mathcal {X}$ over $\mathcal {A}$ , we consider the problem of finding short smooth curves in the sphere $\mathcal {S}_\mathcal {X}$ = x ∈ $\mathcal {X}$ : 〈x, x〉 = 1. Curves in $\mathcal {S}_\mathcal {X}$ are measured considering the Finsler metric which consists of the norm of $\mathcal {X}$ at each tangent space of $\mathcal {S}_\mathcal {X}$ . The initial value problem is solved, for the case when $\mathcal {A}$ is a von Neumann algebra and $\mathcal {X}$ is selfdual: for any element x 0 ∈ $\mathcal {S}_\mathcal {X}$ and any tangent vector ν at x 0, there exists a curve γ(t) = e tZ(x 0), Z ∈ $\mathcal {L}_\mathcal {A} (\mathcal {X})$ , Z* = −Z and ∥Z∥ ≤ π, such that γ(0) = x 0 and $\dot{\gamma }$ (0) = ν, which is minimizing along its path for t ∈ [0, 1]. The existence of such Z is linked to the extension problem of selfadjoint operators. Such minimal curves need not be unique. Also we consider the boundary value problem: given x 0, x 1 ∈ $\mathcal {S}_\mathcal {X}$ , find a curve of minimal length which joins them. We give several partial answers to this question. For instance, let us denote by f 0 the selfadjoint projection I − x 0 ⊗ x 0, if the algebra f 0 $\mathcal {L}_\mathcal {A} (\mathcal {X})$ f 0 is finite dimensional, then there exists a curve γ joining x 0 and x 1, which is minimizing along its path.
LA - eng
KW - C*-modules; spheres; geodesics; -modules
UR - http://eudml.org/doc/269531
ER -

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